JEE Main entrance exam Question Bank Solutions

Let A be a 2 × 2 matrix with det (A) = –1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be ______.

If A^T denotes the transpose of the matrix A = `[(0, 0, a),(0, b, c),(d, e, f)]`, where a, b, c, d, e and f are integers such that abd ≠ 0, then the number of such matrices for which A^–1 = A^T is ______.

The sum of the solutions of the equation `|sqrt(x) - 2|+ sqrt(x)(sqrt(x) - 4) + 2, (x > 0)` is equal to ______.

If the orthocentre of the triangle formed by (1, 3) (4, –5) and (a, b) is (2, 4), Then the value of 33b + 22a is ______.

The sum of the series `tan^-1(1/3) + tan^-1(2/9) + ...... + tan^-1[2^(n-1)/(1 + 2^(2n-1))] + ...... ∞` is ..... `(kπ)/4`. Then the value of k is ______.

If the vertices of a triangle be (0, 0), (6, 0) and (6, 8), then its incentre will be ______.

Sum of roots of the equation (x + 3)² – 4 |x + 3| + 3 = 0 is ______.

The orthocentre of the triangle formed by the lines 4x – 7y + 10 = 0, x + y = 5 and 7x + 4y = 15, is ______.

The coordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2) then the coordinates of its centroid are ______.

If 3x + y = 0 is a tangent to the circle with centre at the point (2, –1), then the equation of the other tangent to the circle from the origin is ______.

If H is the orthocentre of the triangle ABC, then AH is equal to ______.

If α, β, γ, δ are roots of x⁴ – 100x³ + 2x² + 4x + 10 = 0, then `1/α + 1/β + 1/γ + 1/δ` is equal to ______.

Let α, β be the roots of the equation `x^2 - sqrt(2)x + sqrt(6)` = 0 and `1/α^2 + 1, 1/β^2 + 1` be the roots of the equation x² + ax + b = 0. Then the roots of the equation x² – (a + b – 2)x + (a + b + 2) = 0 are ______.

Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?

Let C be the centroid of the triangle with vertices (3, –1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y – 1 = 0 and 3x – y + 1 = 0. Then the line passing through the points C and P also passes through the point ______.

S = `tan^-1(1/(n^2 + n + 1)) + tan^-1(1/(n^2 + 3n + 3)) + ... + tan^-1(1/(1 + (n + 19)(n + 20)))`, then tan S is equal to ______.

The area enclosed between the curve y = log_e(x + e) and the coordinate axes is ______.

The area (in sq.units) of the part of the circle x² + y² = 36, which is outside the parabola y² = 9x, is ______.

The probability of selecting integers a ∈ [–5, 30] such that x² + 2(a + 4)x – 5a + 64 > 0, for all x ∈ R, is ______.

If α and β are the distinct roots of the equation `x^2 + (3)^(1/4)x + 3^(1/2)` = 0, then the value of α⁹⁶(α¹² – 1) + β⁹⁶(β¹² – 1) is equal to ______.